However, to derive the field equations of General Relativity, we extremise the Einstein-Hilbert action

[itex]S_{\text{EH}}[g] = \frac{1}{16 \pi} \int_{\mathcal{M}} d^4x \sqrt{-g} R[/itex] where the integration is performed over the manifold [itex]\mathcal{M}[/itex] and [itex]R[/itex] is the Ricci Scalar

However this suggests that the necessary Lagrangian is

[itex]\mathcal{L}=\frac{1}{16 \pi} R[/itex] i.e. the integrand of the Einstein-Hilbert action.

My question is, why are the two Lagrangians the same. Surely they should be?

Secondly, I understand the motivation for the Einstein-Hilbert action, but where does the first Lagrangian actually come from? As far as I can tell they just seem to introduce it so that we end up with [itex]\ddot{x^\mu} + \Gamma^\mu{}_\nu \rho} \dot{x}^\nu \dot{x}^\rho=0[/itex] after applying the Euler-Lagrange equations.

Why should they be the same ? One is the Lagrangian for the field, the other is for matter, i.e. a particle of mass m.

OK. The field of GR? What does that mean?

From a QFT perspective, I know that particles appear after we quantise the relevant field i.e. an electron is a derived concept after we quantise the classical electron field i.e. give it the necessary anti-commutation relations and what not so that it obeys Fermi-Dirac statistics etc.

But when we talk about field in GR, do we just mean the gravitational field? Would/should(in theory) quantisation of this give rise to the graviton?

I think I'm getting a bit mixed up perhaps because, we can take the field equations [itex]G_{\mu \nu} = T_{\mu \nu}[/itex] and look at a particular component and end up with the equation of motion can't we?

Yes, the field in GR is the gravitational field. One can have matter either as relativistic particles, or by fields, for example fluids or e-m fields. The graviton is the hypothesized quanta of the gravitational field.

Yes, the field in GR is the gravitational field. One can have matter either as relativistic particles, or by fields, for example fluids or e-m fields. The graviton is the hypothesized quanta of the gravitational field.

so if we were using the gravitational field then matter would move as a relativistic (timelike) particle in the gravitational field. Is it true that we can get it's equation of motion by examining particular components of the field equation?

Surely this is true? For example, looking at the tt component of the field equation will tell us how time will dilate for relativistic particles moving through the gravitational field, yes?